# Scalar QCD

1. ### abrata

1
Hi all,

I am currently trying to calculate the beta function for scalar QCD theory (one loop for general su(n)).

I therefore need to calculate the Feynman rules in order to apply them to the one loop diagrams. Unfortunately I am getting very confused with what the Lagrangian for scalar QCD should be. If anyone knows of some clear examples of this Lagrangian and possibly the derivation of the corresponding Feynman rules and diagrams I would be very appreciated.

Many thanks
Abrata

2. ### The_Duck

916
First start with the Lagrangian for a scalar field with an internal SU(N) symmetry:

$$(\partial_\mu \phi^a)^\dagger (\partial^\mu \phi^a) - m^2 \phi^{a \dagger} \phi^a - \frac{\lambda}{4}(\phi^{a\dagger} \phi^a)^2$$

Then replace the partial derivatives with covariant derivatives:

$$(D_\mu \phi^a)^\dagger (D^\mu \phi^a) - m^2 \phi^{a \dagger} \phi^a - \frac{\lambda}{4}(\phi^{a\dagger} \phi^a)^2$$

where ##D_\mu \phi^a = \partial_\mu \phi^a - i g T^{a b} \phi^b##. Add in the gauge field Lagrangian and you have the Lagrangian for scalar QCD.

Srednicki's textbook has some chapters on scalar electrodynamics, which might help you.